1888. ] NEW YORK ACADEMY OF SCIENCES. 239 
For the investigation of quality we need a method of represent- 
ing musical sounds graphically before a path of research is 
opened. (The speaker then gave illustrations of the application 
of the graphic method, analyzing each case, and dwelling more 
particularly on Koenig’s mode of compounding the parallel vibra- 
tions of two forks.) 
Assume a compound musical sound made up of six com- 
ponents whose wave lengths form the series 1, 4, 4,4, 4, 4. 
Draw all their sinusoids on the same axis, and compound these 
into a single curve, remembering that each loop with its con- 
vexity upward represents a momentary slight increase of inten- 
sity, and each with its convexity downward a momentary slight 
decrease of intensity. Then, as each compound wave strikes 
the ear, there are six subordinate impressions along with the 
general impression of a push and a pull on the auditory 
medium. ‘These occur in a definite order, and the groups are 
repeated periodically. The result is a characteristic sensation 
quite different from that which arises if the fandamental sound 
alone impresses the ear. The quality is more brilliant, while the 
fundamental pitch remains the same. It might be quite natural 
to assume, as was once commonly done, that quality must, in 
some way, depend on the form of the sound wave. It was re- 
served tor Helmholtz to give to this idea a firm experimental 
basis, and to show that quality depends upon the number, 
orders, and relative intensities of the upper partial tones which 
accompany the fundamental. His demonstration was a verifica- 
tion of what had previously been only guessed; for, if we vary 
either the number, or the orders, or the relative intensities of 
these upper partials, and plot the resultant curves, the latter 
become changed in form with the change in quality. 
But there is another way, independently of those just men- 
tioned, in which the wave form may be varied. It is that of 
varying the mode of combination of the same set of components. 
The accompanying figure, due to Koenig, shows groups of 
sinusoids compounded together with variations of phase between 
the fundamental and the upper partials. In the group marked 0, 
all begin together at the same origin, and hence with perfect 
coincidence of phase. In that marked 4, there is a difference in 
phase of a fourth of a wave length of the fundamental between 
this and its first upper partial. In those marked 4 and 3, respec- 
tively, this difference of phase is a half-wave length and three- 
fourths of a wave length of the fundamental, respectively. In 
figures a and 0, the amplitudes form a harmonic series de- 
termined by the corresponding wave lengths, a@ including the 
full series, and 4 the odd components only. In figures ¢ and d, 
the amplitudes form a decreasing geometrical series determined 
